ON THE MOTION OF PENDULUMS. 133 importance. Making this simplification and substituting in (191) we get where c= If then J.0 be the initial and A the final value of Ati we get from (192) ...(193). Let now J.0 -h AJ.0 be what J.0 would become if, while the final arc A and the whole time t remained the same, the motion had been going on for an indefinite time before the epoch from which t is measured, in which case the superior limit in the integral involved in the expression for 6r would have been oo in place of t. Then q = c f * jsin nt r n(t_ } dt^i dtf $ > Jo ( Jo v&J whence by subtracting, member from member, equation (193) from equation (194), we get log -o-o -m nt fcos n (t _ 4 ) o I J^ which becomes after integration by parts + , lo c TT ft . ^ _ ^ = j / -- 2^ • cos nt - cos 2n£ cos nt -7- Jt */t 8m2nt)[ sin w* ^[ ...... (195). Now it is supposed to be very large : in Coulomb's experiments in fact 10 oscillations were observed, so that nt = lOyr. But when t is at all large the two integrals r dt r - , dt cos nt -77, sin ?££ — 7^ v^ Jt vt can be expressed under the forms — P sin nt -f Q cos w£, P cos 7^ -f Q sin n^, where ll •' i ^» 1 .3.5.